Stochastic theory of minimal realization

This paper exploits the concept of a predictor space in the minimal realization problem for systems generating an analytic impulse response matrix. The predictor space constructed, by stochastic input and output processes forms the state space for the stochastic system representation, where a system is represented by the basis of the predictor space and the innovation process of input. The minimal realization problem is then solved for a given analytic impulse response matrix by defining a stochastic system driven by white noise whose input-output covariance equals the given impulse response matrix. It is shown that the coefficient matrices of the stochastic system representation constitute a solution to the minimal realization problem for the deterministic system with given impulse response matrix. The paper provides a unifying overview to many aspects of the realization problem and its algorithms.